Given $ m \angle CBD = 6x + 27$, $ m \angle ABC = 7x - 9$, and $ m \angle ABD = 96$, find $m\angle CBD$. $B$ $A$ $D$ $C$
From the diagram, we see that together ${\angle ABC}$ and ${\angle CBD}$ form ${\angle ABD}$ , so $ {m\angle ABC} + {m\angle CBD} = {m\angle ABD}$ Substitute in the expressions that were given for each measure: $ {7x - 9} + {6x + 27} = {96}$ Combine like terms: $ 13x + 18 = 96$ Subtract $18$ from both sides: $ 13x = 78$ Divide both sides by $13$ to find $x$ $ x = 6$ Substitute $6$ for $x$ in the expression that was given for $m\angle CBD$ $ m\angle CBD = 6({6}) + 27$ Simplify: $ {m\angle CBD = 36 + 27}$ So ${m\angle CBD = 63}$.